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Dynamics Forced by Homoclinic Orbits

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Dynamics Forced by Homoclinic Orbits DYNAMICS FORCED BY HOMOCLINIC ORBITS VALENT´IN MENDOZA Abstract. The complexity of a dynamical system exhibiting a homoclinic orbit is given by the orbits that it forces. In this work we present a method, based in pruning theory, to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the 2-disk. Due to Cantwell and Conlon, this set is uniquely determined in the isotopy class of the orbit, up a topological conjugacy, so it contains the dynamics forced by the homoclinic orbit. Moreover we apply the method for finding the orbits forced by certain infinite families of homoclinic horseshoe orbits and propose its generalization to an arbitrary Smale map. 1. Introduction Since the Poincar´e'sdiscovery of homoclinic orbits, it is known that dynamical systems with one of these orbits have a very complex behaviour. Such a feature was explained by Smale in terms of his celebrated horseshoe map [40]; more precisely, if a surface diffeomorphism f has a homoclinic point then, there exists an invariant set Λ where a power of f is conjugated to the shift σ defined on the compact space Σ2 = f0; 1gZ of symbol sequences. To understand how complex is a diffeomorphism having a periodic or homoclinic orbit, we need the notion of forcing. First we will define it for periodic orbits. 1.1. Braid types of periodic orbits. Let P and Q be two periodic orbits of homeomorphisms f and g of the closed disk D2, respectively. We say that (P; f) and (Q; g) are equivalent if there is an orientation-preserving homeomorphism h : D2 ! D2 with h(P ) = Q such that f is isotopic to h−1 ◦ g ◦ h relative to P . The equivalence class containing (P; f) is called the braid type bt(P; f). When the homeomorphism f is fixed, it will be written bt(P ) instead of bt(P; f). Now we can define the forcing relation between two braid types β and γ. We say that β forces γ, 2 denoted by β >2 γ, if every homeomorphism of D which has an orbit with braid type β, exhibits also an orbit with braid type γ. So we say that P forces Q, denoted by P >2 Q, if bt(P ) >2 bt(Q). In [6] Boyland proved that >2 is a partial order. If E(P ) is the set of periodic points whose orbits are forced by P , one can ensure that the dynamics of a diffeomorphism f containing an orbit with the braid arXiv:1412.2737v5 [math.DS] 23 Apr 2019 type of P is at least as complicated as f is restricted to E(P ). To find E(P ) it is necessary to use the Nielsen-Thurston theory of classification of surface homeomorphisms up to isotopies. In fact, Asimov and Franks [3] and Hall [25] showed that if f is pseudo-Anosov on D2 n P then E(P ) is formed by the periodic points of its canonical representative φ. If P is of reducible type then the Nielsen-Thurston representative φ does not have the minimal number of periodic orbits, but a refinement of φ, due to Boyland [6,7], is used to get a condensed map which satisfies that property. An interesting case is when P is a periodic orbit of a Smale map f on D2, that is, f is an Axiom A map with a strong transversality between the invariant manifolds of its non-wandering points. In this Date: October 30, 2015. 2010 Mathematics Subject Classification. Primary 37E30, 37E15, 37C29. Secondary 37B10, 37D20. Key words and phrases. Homoclinic orbits, braid types, dynamical core, Smale horseshoe, pruning theory. 1 2 VALENT´IN MENDOZA context the notion of bigon has became a tool for finding the Nielsen-Thurston representative rel to P : a bigon is a simply connected open region disjoint from P which is bounded by a stable segment and an unstable segment. In fact, Bonatti and Jeandenans [5] have proved that if there are no wandering bigons rel to a periodic orbit P of a transitive basic piece K, there exists a pseudo-Anosov map φ, with possibly 1-pronged singularities, and a semiconjugacy π between f and φ which is injective on the set of periodic orbits of K except on the boundary ones. If we do not admit non-wandering bigons too, the Bonatti-Jeandenans map φ is actually pseudo-Anosov and then the non-boundary periodic orbits of K are all forced by P . Thus the non-existence of bigons implies that the periodic orbits of K are forced by P . A similar result was obtained by Lewowicz and Ures in [33]: if f is a Smale map without bigons relative to P , which is called exteriorly situated, then its basic set is contained in the persistent set given by Handel [27] relative to P . More precisely it follows from Grines [23, 22] that if the non-wandering set of a Smale map contains an exteriorly situated basic set that does not contain special pairs of boundary periodic point (definition6) then there exists a semiconjugacy between f and a hyperbolic homeomorphism f0 which can be considered the canonical hyperbolic representative of the homotopy class of f. See [2, 24]. Since there exists a decomposition in basic pieces for Smale maps, the same analysis can even be applied if f has several transitive pieces and does not have bigons rel to P , considering the return maps to each basic piece. It will be the case, for example, if P is a renormalization of two or more orbits of pseudo-Anosov type [35]. 1.2. Forcing on homoclinic orbits. Let us now suppose that P is a homoclinic orbit to a fixed point of a Smale diffeomorphism. In that case there are substantial differences but also useful similarities with the periodic case. First in [34] Los has proved that the forcing relation on periodic orbits can be extended to homoclinic or heteroclinic orbits in a suitable topology. There are several methods for finding the dynamics forced by a homoclinic orbit of a diffeomorphism f. For example, in [11, 12], Collins has constructed surface hyperbolic diffeomorphisms associated to a homoclinic orbit which can be used for approximating the entropy of that orbit as close as we want. His method uses the homoclinic tangle associated to the orbit. In [41, 43], Yamaguchi and Tanikawa have studied the forcing relation of reversible homoclinic horseshoe orbits appearing in area-preserving H´enonmaps. In other direction Boyland and Hall have given conditions for which a periodic orbit is isotopy stable relative to a compact set [8], and their result can be used for studying homoclinic orbits. 2 Since f restricted to MP := Int(D ) n P is isotopic to an end periodic homeomorphism (lemma8), it is more convenient to use the Handel-Miller theory of classification of end periodic automorphisms on surfaces [29] as is presented in [20]. In this case Cantwell and Conlon [10] have proved that there exist a Handel-Miller map h and a set CP , called dynamical core, which is the intersection of the pair of totally disconnected and transverse geodesic laminations, such that h: C!C is uniquely determined by the isotopy class of f on MP . Thus one can say as definition that CP is the set of orbits forced by P or that the dynamics of h on CP is forced by P : we say that an (finite or infinite) h-orbit Q is forced by P , denoted by P >2 Q, if Q ⊂ CP . In the general case, if an end periodic map f is irreducible [20], the union of the laminations fills M and f is isotopic to a pseudo-Anosov-like representative of the dynamics rel to P which preserves a pair of geodesic laminations. In our case, as in the work of Grines [23] for surfaces of finite topology, one has the following result (theorem 14). DYNAMICS FORCED BY HOMOCLINIC ORBITS 3 Theorem 1. Let f be a Smale map with a exteriorly situated basic set K without special pairs rel to an homoclinic orbit P . Then CP = K up a topological finite-to-one semi-conjugacy ι : CP ! K which is injective on the set of non-boundary periodic points. Therefore, in the hypothesis of theorem1, K contains the orbits forced by P . 1.3. Results of the paper. This is the context where our work is inserted. An important element of this paper is the pruning theory introduced by de Carvalho in [14] which is a technique for eliminating dynamics of a surface homeomorphism. So in section2 we state the differentiable pruning theorem (theorem 18), due to de Carvalho and the author, that can be used to eliminate bigons of a Smale map and hence for finding the dynamical core associate to a homoclinic orbit. This differentiable version of pruning have been implemented for uncrossing invariant manifolds of a Smale diffeomorphism in a region called a pruning domain. This point of view was inspired by the work of P. Cvitanovi´c[13] where a generic one-folding map is interpreted as a partial horseshoe, that is, a map whose dynamics forbids or prunes certain horseshoe orbits. As an application we will study, in section3, the forcing relation of homoclinic orbits coming from the standard Smale horseshoe F stated in [40]. In particular, it will be dealt homoclinic orbits to the 1 w 1 1 1 1 fixed point 0 , that is, orbits P0 whose code in Σ2 is 010w0 · 10 where w is a finite word called w decoration.
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